Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| Index: | $2688$ | $\PSL_2$-index: | $1344$ | ||||
| Genus: | $97 = 1 + \frac{ 1344 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 32 }{2}$ | ||||||
| Cusps: | $32$ (of which $2$ are rational) | Cusp widths | $28^{16}\cdot56^{16}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot3^{2}\cdot6^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $12$ | ||||||
| $\Q$-gonality: | $14 \le \gamma \le 48$ | ||||||
| $\overline{\Q}$-gonality: | $14 \le \gamma \le 48$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.2688.97.146 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}15&28\\44&11\end{bmatrix}$, $\begin{bmatrix}17&36\\2&25\end{bmatrix}$, $\begin{bmatrix}17&36\\46&25\end{bmatrix}$, $\begin{bmatrix}23&24\\30&5\end{bmatrix}$, $\begin{bmatrix}41&16\\40&15\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.1344.97.ee.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $4$ |
| Cyclic 56-torsion field degree: | $96$ |
| Full 56-torsion field degree: | $1152$ |
Jacobian
| Conductor: | $2^{399}\cdot7^{167}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{13}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| Newforms: | 14.2.a.a$^{3}$, 56.2.a.a, 56.2.a.b, 56.2.b.a, 56.2.b.b, 64.2.a.a$^{2}$, 98.2.a.b$^{3}$, 196.2.a.b$^{2}$, 196.2.a.c$^{2}$, 224.2.b.a, 224.2.b.b, 392.2.a.c, 392.2.a.f, 392.2.a.g, 392.2.b.e, 392.2.b.g, 448.2.a.b, 448.2.a.f, 448.2.a.i, 448.2.a.j, 1568.2.b.f, 1568.2.b.g, 3136.2.a.bd, 3136.2.a.bh, 3136.2.a.bi, 3136.2.a.bj, 3136.2.a.bl, 3136.2.a.bo, 3136.2.a.bv, 3136.2.a.bw, 3136.2.a.bz |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(7)$ | $7$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 8.96.1-8.m.2.6 | $8$ | $28$ | $28$ | $1$ | $0$ | $1^{12}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.m.2.6 | $8$ | $28$ | $28$ | $1$ | $0$ | $1^{12}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.1344.45-56.u.1.3 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{4}\cdot2^{11}\cdot4^{2}\cdot6\cdot12$ |
| 56.1344.45-56.u.1.47 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{4}\cdot2^{11}\cdot4^{2}\cdot6\cdot12$ |
| 56.1344.45-56.bb.1.4 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{4}\cdot2^{11}\cdot4^{2}\cdot6\cdot12$ |
| 56.1344.45-56.bb.1.48 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{4}\cdot2^{11}\cdot4^{2}\cdot6\cdot12$ |
| 56.1344.49-56.d.1.8 | $56$ | $2$ | $2$ | $49$ | $12$ | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.1344.49-56.d.1.19 | $56$ | $2$ | $2$ | $49$ | $12$ | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.5376.193-56.hh.2.17 | $56$ | $2$ | $2$ | $193$ | $30$ | $1^{30}\cdot2^{15}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.hw.2.2 | $56$ | $2$ | $2$ | $193$ | $38$ | $1^{30}\cdot2^{15}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.kb.2.17 | $56$ | $2$ | $2$ | $193$ | $28$ | $1^{30}\cdot2^{15}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.kq.1.5 | $56$ | $2$ | $2$ | $193$ | $34$ | $1^{30}\cdot2^{15}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.lc.2.8 | $56$ | $2$ | $2$ | $193$ | $29$ | $1^{24}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.px.2.4 | $56$ | $2$ | $2$ | $193$ | $29$ | $1^{24}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.qo.2.8 | $56$ | $2$ | $2$ | $193$ | $31$ | $1^{24}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.qy.2.10 | $56$ | $2$ | $2$ | $193$ | $31$ | $1^{24}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.rg.1.7 | $56$ | $2$ | $2$ | $193$ | $30$ | $1^{40}\cdot2^{10}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.rm.2.8 | $56$ | $2$ | $2$ | $193$ | $30$ | $1^{40}\cdot2^{10}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.ry.1.6 | $56$ | $2$ | $2$ | $193$ | $30$ | $1^{40}\cdot2^{10}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.sa.1.8 | $56$ | $2$ | $2$ | $193$ | $30$ | $1^{40}\cdot2^{10}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.so.2.8 | $56$ | $2$ | $2$ | $193$ | $30$ | $1^{40}\cdot2^{10}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.sq.1.6 | $56$ | $2$ | $2$ | $193$ | $30$ | $1^{40}\cdot2^{10}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.tc.1.8 | $56$ | $2$ | $2$ | $193$ | $30$ | $1^{40}\cdot2^{10}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.ti.1.4 | $56$ | $2$ | $2$ | $193$ | $30$ | $1^{40}\cdot2^{10}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.tq.1.3 | $56$ | $2$ | $2$ | $193$ | $31$ | $1^{24}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.ua.1.4 | $56$ | $2$ | $2$ | $193$ | $31$ | $1^{24}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.ue.2.11 | $56$ | $2$ | $2$ | $193$ | $29$ | $1^{24}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.ul.2.8 | $56$ | $2$ | $2$ | $193$ | $29$ | $1^{24}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.wn.1.6 | $56$ | $2$ | $2$ | $193$ | $36$ | $1^{30}\cdot2^{15}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.wy.1.7 | $56$ | $2$ | $2$ | $193$ | $27$ | $1^{30}\cdot2^{15}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.yr.1.7 | $56$ | $2$ | $2$ | $193$ | $36$ | $1^{30}\cdot2^{15}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.zc.1.4 | $56$ | $2$ | $2$ | $193$ | $23$ | $1^{30}\cdot2^{15}\cdot4^{6}\cdot6^{2}$ |
| 56.8064.289-56.ua.1.10 | $56$ | $3$ | $3$ | $289$ | $44$ | $1^{38}\cdot2^{33}\cdot4^{7}\cdot6^{6}\cdot12^{2}$ |